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In the figure shown above, if the are of the shaded region

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In the figure shown above, if the are of the shaded region

by swerve » Wed Aug 14, 2019 10:29 am

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In the figure shown above, if the area of the shaded region is 3 times the area of the smaller circular region, then the circumference of the larger circle is how many times the circumference of the smaller circle?

A. 4
B. 3
C. 2
D. \(\sqrt{3}\)
E. \(\sqrt{2}\)

The OA is C

Source: Official Guide
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by GMATGuruNY » Wed Aug 14, 2019 10:48 am
swerve wrote:Image

In the figure shown above, if the area of the shaded region is 3 times the area of the smaller circular region, then the circumference of the larger circle is how many times the circumference of the smaller circle?

A. 4
B. 3
C. 2
D. \(\sqrt{3}\)
E. \(\sqrt{2}\)
Smaller circle:
Let r=1.
Area = πr² = π(1²) = π
Circumference = 2Ï€r = 2Ï€(1) = 2Ï€

Since the area of the shaded region is 3 times the area of the smaller circle, the shaded region = 3Ï€.

Larger circle:
Area = (smaller circle) + (shaded region) = π + 3π = 4π.
Thus:
πr² = 4π
r = 2
Circumference = 2Ï€r = 2Ï€(2) = 4Ï€

The circumference of the larger circle is how many times the circumference of the smaller circle?
(larger circumference)/(smaller circumference) = 4Ï€/2Ï€ = 2

The correct answer is C.
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by Scott@TargetTestPrep » Sun Aug 18, 2019 6:29 pm
swerve wrote:Image

In the figure shown above, if the area of the shaded region is 3 times the area of the smaller circular region, then the circumference of the larger circle is how many times the circumference of the smaller circle?

A. 4
B. 3
C. 2
D. \(\sqrt{3}\)
E. \(\sqrt{2}\)

The OA is C

Source: Official Guide
We can let a = the radius of the entire figure (i.e, the larger circle), and b = the radius of the whtie circle (i.e., the smaller circle), and we have:

π(a^2 - b^2) = 3 * πb^2

a^2 - b^2 = 3b^2

a^2 = 4b^2

a = 2b

The circumference of the larger circle is 2Ï€a = 4Ï€b and the circumference of the smaller circle is 2Ï€b. Therefore, the circumference of the larger circle is twice that of the smaller circle.

Answer: C

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